The likelihood ratio test for the change point problem for exponentially distributed random variables. (English) Zbl 0671.62019
Let \(x_ 1,x_ 2,...,x_ i,...,x_{n+1}\) be \(n+1\) independent and exponentially distributed random variables with intensity \(\lambda_ 1\) for \(i\leq \tau\) and \(\lambda_ 2\) for \(i>\tau\), where \(\tau\) as well as \(\lambda_ 1\) and \(\lambda_ 2\) are unknown. The authors consider the likelihood ratio (LR) statistic for testing \(H_ 0:\) \(\lambda_ 1=\lambda_ 2\) versus \(H_ 1:\) \(\lambda_ 1\neq \lambda_ 2\), and show that the asymptotic null-distribution of the LR statistic is an extreme value distribution.
It is also shown that the test is optimal in the sense of Bahadur although the Pitman efficiency is zero. Furthermore, simulation shows good power for those values of n which are relevant to most applications \((20<n<200)\).
It is also shown that the test is optimal in the sense of Bahadur although the Pitman efficiency is zero. Furthermore, simulation shows good power for those values of n which are relevant to most applications \((20<n<200)\).
Reviewer: R.A.Khan
Keywords:
Bahadur efficiency; change point problem; exponential distribution; likelihood ratio test; normed uniform quantile process; power; properties; asymptotic null-distribution; extreme value distribution; Pitman efficiency; simulationReferences:
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